Probabilistic flood hazard mapping- effects of uncertain boundary conditions
Hydrol.Earth Syst.Sci.,17,3127–3140,https://www.wendangku.net/doc/fa5632203.html,/17/3127/2013/doi:10.5194/hess-17-3127-2013
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Probabilistic ?ood hazard mapping:effects of uncertain boundary conditions
A.Domeneghetti 1,S.Vorogushyn 2,A.Castellarin 1,
B.Merz 2,and A.Brath 1
1School
of Civil Engineering,Department DICAM,University of Bologna,Bologna,Italy
2GFZ German Research Centre for Geosciences,Section 5.4:Hydrology,Potsdam,Germany Correspondence to:A.Domeneghetti (alessio.domeneghetti@unibo.it)
Received:25July 2012–Published in Hydrol.Earth Syst.Sci.Discuss.:29August 2012Revised:24June 2013–Accepted:3July 2013–Published:5August https://www.wendangku.net/doc/fa5632203.html,prehensive ?ood risk assessment studies should quantify the global uncertainty in ?ood hazard es-timation,for instance by mapping inundation extents to-gether with their con?dence intervals.This appears of par-ticular importance in the case of ?ood hazard assessments along dike-protected reaches,where the possibility of oc-currence of dike failures may considerably enhance the un-certainty.We present a methodology to derive probabilistic ?ood maps in dike-protected ?ood prone areas,where sev-eral sources of uncertainty are taken into account.In particu-lar,this paper focuses on a 50km reach of River Po (Italy)and three major sources of uncertainty in hydraulic mod-elling and ?ood mapping:uncertainties in the (i)upstream and (ii)downstream boundary conditions,and (iii)uncer-tainties in dike failures.Uncertainties in the de?nition of upstream boundary conditions (i.e.design-hydrographs)are assessed through a copula-based bivariate analysis of ?ood peaks and volumes.Uncertainties in the de?nition of down-stream boundary conditions are characterised by uncertainty in the rating curve with con?dence intervals which re?ect discharge measurement and interpolation errors.The effects of uncertainties in boundary conditions and randomness of dike failures are assessed by means of the Inundation Haz-ard Assessment Model (IHAM),a recently proposed hybrid probabilistic-deterministic model that considers three differ-ent dike failure mechanisms:overtopping,piping and micro-instability due to seepage.The results of the study show that the IHAM-based analysis enables probabilistic ?ood hazard mapping and provides decision-makers with a fundamental piece of information for devising and implementing ?ood risk mitigation strategies in the presence of various sources of uncertainty.
1
Introduction
Many studies in the literature highlight how inundation haz-ard and risk assessments are affected by several sources of uncertainties which limit their reliability (e.g.Merz and Thieken,2005;Apel et al.,2004,2008;Most and Wehrung,2005;Hall and Solomatine,2008).In this context,there is a consensus in the scienti?c community that a proper risk anal-ysis should provide an indication of uncertainty,emphasising how the identi?cation of the optimal ?ood risk management strategy can be pursued only if all major sources of uncer-tainty are adequately taken into consideration and a quanti?-cation of their impacts is provided (USACE,1992).
Uncertainty has always been inherent in ?ood assessment and considered in ?ood defence engineering by means for example of adoption of an adequate freeboard (Hall and Solomatine,2008).The unavoidable presence of uncertainty can be attributed to the fact that ?ood risk evaluations are usually carried out for extreme events that are seldom ob-served,which makes the calibration of ?ood risk assessment models dif?cult,if not impossible (Apel et al.,2004).Un-der such circumstances,the evaluation of uncertainty sources is a pragmatic extension to conventional validation.Further-more,Hall and Solomatine (2008)and Apel et al.(2008)emphasise this need,highlighting how the quanti?cation of the uncertainty could help to judge the consistency and the reliability of hydraulic risk assessment as well as to pro-vide useful advices for future data collection or research activities in order to yield more reliable results.In a con-text where model calibration and validation is dif?cult due to consideration of extreme events or lack of data,Hall and Anderson (2002)and Hall (2003)suggest a transparent and
Published by Copernicus Publications on behalf of the European Geosciences Union.
comprehensive description of the cause-effect relationships adopted in the methodology and implemented in mathemat-ical formulations.This is particularly relevant in the case of dike failure analysis,where the uniqueness of breaches re-duces or even eliminates the possibility to calibrate and val-idate deterministic numerical models.Evaluation of possible scenarios could only be handled by means of causal mod-els considering uncertainties in dike breach processes(Hall, 2003;V orogushyn et al.,2010).
In practical applications,the assessment of inundation ar-eas is usually carried out in a deterministic fashion by means of hydraulic models.Those are?rst calibrated relative to a speci?c historical?ood event,and then used to estimate ?ood extents relative to different(and typically higher)event magnitudes.This procedure,even when physically based and numerically complex models are considered(e.g.fully2-D model,etc.),relies on some fundamental assumptions that may be summarised as follows:(i)capability of the model to correctly reproduce the hydraulic behaviour of the river and inundated?oodplains;(ii)time stationarity of model pa-rameters,i.e.the roughness coef?cients calibrated for a spe-ci?c event are considered suitable for a range of?ooding scenarios that could differ signi?cantly from the calibration event;(iii)all hydraulic information(i.e.?ow hydrographs, rating curves)are error-free.
In a context characterised by these sources of uncer-tainty,the de?nition of probabilistic?ood hazard and?ood risk maps appear the most reasonable way to proceed.Di Baldassarre(2012)argues that there are at least three main reasons why probabilistic?ood hazard maps should be pre-ferred to deterministic ones:(1)hydrological and hydraulic analysis are always affected by uncertainty,which often can-not be neglected;(2)a fair presentation of the results of any analysis should also quantify and illustrate the associated un-certainty,and this can be accomplished only in a probabilis-tic framework;(3)stakeholders and decision-makers should be provided by hydrologists with probabilistic inundation maps to guide and support the de?nition of?ood mitiga-tion strategies;when deterministic maps are produced it im-plies that a decision has already been made by hydrologists, who are hence no longer behaving like scientists,but rather as decision-makers themselves.As a result,the deterministic estimation of?ood extension may involve inexact and dan-gerous consequences,especially if it is used for planning and development purpose in the?ood-prone area.In?ood risk research,a number of studies have already considered and classi?ed various uncertainty sources based on the distinc-tion between two types of uncertainty:(i)natural or aleatory uncertainty,associated with the natural variability of the phe-nomena of interest and(ii)epistemic uncertainty,resulting from imperfect knowledge of the system(e.g.Apel et al., 2004;Hall and Solomatine,2008;Merz and Thieken,2005; Most and Wehrung,2005),or from simpli?cations associated with the selected modelling approach and parametrizations (e.g.1-D model instead of2-D,constant or distributed rough-ness coef?cients etc.).
Many previous studies analysed the effect of uncer-tainty associated with roughness parametrizations of hy-draulic models(Aronica et al.,2002;Bates et al.,2004; Pappenberger et al.,2005).Additionally,Pappenberger et al. (2006)analysed the uncertainty in upstream and downstream boundary conditions when applied to?ood inundation pre-dictions with a1-D?ow model.Other authors considered additional uncertainties in?ood hazard and risk chain,in-cluding extreme value statistics(Apel et al.,2008;Merz and Thieken,2009),dike breach processes,e.g.breach locations and dimension(Apel et al.,2004;V orogushyn et al.,2010, 2011)as well as?ood damage estimations(Apel et al.,2008; Merz and Thieken,2009;de Moel et al.,2011;V orogushyn et al.,2012).They concluded that currently uncertainties in damage estimations and in extreme value statistics dominate the uncertainties in risk estimates,although this conclusion remains site-speci?c.
The effects of uncertain(upstream and downstream) boundary conditions on?ood hazard assessment is still poorly understood,and the literature on this topic is sparse. Our analysis focuses in particular on the uncertainty associ-ated with rating curves used as downstream boundary condi-tions,while the aleatory uncertainty related to the selection of a design hydrograph is taken into account,referring to dif-ferent?ood hydrographs estimated with a bivariate?ood fre-quency analysis.
The effect of the downstream boundary condition on the area of interest is reduced,if not completely removed,by ex-tending the hydraulic model far downstream of the area of interest.However,this expedient may be costly and time con-suming to implement,or dif?cult due to a lack of data.To ad-dress these issues the modeller needs to consider if and how the uncertainty in the downstream boundary condition im-pacts her/his computations.Since the effects of rating-curve uncertainty on?ood hazard mapping is the main goal of our investigation,we deliberately referred to a case in which we set the boundary condition at the downstream end of the con-sidered river reach.
Even though the literature reports several studies high-lighting the global uncertainty affecting discharge mea-surements and rating-curve construction(e.g.Domeneghetti et al.,2012;Di Baldassarre and Claps,2011;Di Baldassarre and Montanari,2009),the literature on the effects of rating-curves uncertainty of?ood hazard and?ood risk assessments is still sparse.Moreover,institutions and agencies in charge of hydroclimatic monitoring usually do not provide prac-titioners and users with indications of uncertainty associ-ated with rating curves.Conversely,rating curves are usually utilised in a deterministic way although their sampling vari-ability may be signi?cant and may play a dominant role in practical applications(Domeneghetti et al.,2012).
Our study makes use of the outcomes of a previous analy-sis on rating-curve uncertainty performed for the same river
Hydrol.Earth Syst.Sci.,17,3127–3140,https://www.wendangku.net/doc/fa5632203.html,/17/3127/2013/
Fig.1.system
area (yellow).
reach
pact of
ping.Our investigation was performed by setting up a hybrid probabilistic-deterministic?ood hazard assessment model for the?ood-prone areas located along a diked reach of the lower portion of the Po River.We discuss how the considera-tion of this uncertainty may impact?ood management deci-sions compared to a deterministic speci?cation of boundary conditions.
2Methodology
Chains of models that describe?uvial inundation processes and?ood damages are typically applied for?ood hazard and risk assessment.In this approach,each modelling step or chain link exhibits a number of inherent uncertainties that are summarised in Table1,starting from a triggering event to the?nal inundation pattern.Referring to some natural and epistemic sources of uncertainty(sources listed in italic in Table1),the study aims at quantifying the contribution of different terms of uncertainty,evaluating the feasibility and the amount of uncertainty reduction that can be achieved by adopting additional information or different procedure.We analyse the role of uncertain boundary conditions on?ood hazard statements by means of the Inundation Hazard As-sessment Model(IHAM)(V orogushyn et al.,2010).
IHAM model is a hybrid probabilistic-deterministic model developed for?ood hazard assessment along protected river reaches considering dike failures.The model is comprised of three main modules:an unsteady one-dimensional hydraulic
dikes,
dike sys-tem stability under hydraulic load conditions,and a2-D raster-based diffusive wave model(2-D raster-based model; Merz,1996)for the simulation of?oodplain?ow in the case of dike failures(see Fig.2).All three modules are contin-uously coupled at runtime.The1-D model routes a?ood wave in the river channel and over?oodplains between dikes. It computes the hydraulic load on?ood protection dikes in terms of water level and impoundment duration.During the simulation,each discretised dike section is evaluated for fail-ure due to overtopping,piping and slope instability due to seepage?ow through the embankment(micro-instability;see V orogushyn et al.,2009).In the case of dike failure,the out-?ow volume through the breach into the?ood-prone area is computed and used as a boundary condition in the2-D stor-age cell model.The simulation of water exchange between river channel and?oodplain,including the reverse?ow,is in-corporated by means of a continuous data exchange between modules.A distinctive characteristic of the IHAM model is the coupled modelling chain of channel?ow,dike failure and inundation processes without a priori assumption on the lo-cation,time and characteristics of the dike failure.Those are determined during the simulation based on the current hy-draulic load and dike propensity to failure.
The schematic structure of the IHAM model is shown in Fig.2,which highlights the model core system(three coupled modules),and the pre-(input)and post-(output) processor phases.The modelling system is run in a Monte Carlo framework(MC)to address the considered sources of
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Table 1.Sources of uncertainty in ?ood hazard mapping grouped into natural and epistemic uncertainty (adapted from Apel et al.,2004);sources in italic are directly considered into the presented analysis.
Modules
Natural uncertainty
Epistemic uncertainty –measurement error;
–limited time series length;(1)Hydrological analysis
–annual maximum discharge;–statistical inference;–?ow hydrograph shape ;
–parameter estimation
–peak discharge estimation;–?ow hydrograph wave form;
–discharge measurement errors;
–mathematical expression for rating-curve estimation;(2)Rating-curve -variation of river geometry in time;
–number of pair used for rating-curve estimation;–methodology for rating-curve estimation;–interpolation/extrapolation errors;–error in model selection;(3)Flood routing –variation of river geometry over time;
–numerical simpli?cation;–parameter calibration;
–geometrical variation over space;
–measurements errors of levee geometry;(4)Dike stability
–variation of geotechnical parameters in space;
–variability estimations of levee parameters
–?nal width and development time of levee breaches;
(permeability,turf quality,material cohesion,etc.);–formalisation of dike breach processes;–error in model selection;(5)Flood dynamics
–variability of surface roughness in –numerical simpli?cation;space and time due to variable land use;
–DEM inaccuracy;–parameter
estimation;
Fig.2.Schematic structure of the IHAM model adopted for ?ood hazard estimation under uncertainty conditions.
uncertainty (e.g.upstream and downstream boundary condi-tions)and the stochasticity of dike breaching processes.IHAM model considers the uncertainty related to dike sys-tem stability implementing the “Dike breach module”(see Fig.2and Sect.3.2).It evaluates the probability of dike fail-ures upon hydraulic loading computed by the 1-D model.Each section of the dike system with a length of approxi-mately 1.2km is tested for dike stability based on the current load during the whole simulation.The probability of fail-ure for a given hydraulic load is estimated through fragility curves (see e.g.Sayers et al.,2002)de?ned for each dike sec-tion for three failure mechanisms:overtopping,piping and micro-instability (Apel et al.,2004;V orogushyn et al.,2009).In the case of single or multiple dike collapses,the de-velopment time and the ?nal dimension of each breach are stochastically generated based on probability distribu-tion functions ?tted to historical observations (see Govi and Turitto,2000,and Sect.3.2).
Limited knowledge about ?ow dynamics,errors on ?ow-rates measurements and inaccuracy related to the applied methodology for rating-curve estimation (epistemic uncer-tainties)are considered in an MC simulation.As a result,the IHAM model computes dike failure probabilities for the whole embankment system and provides probabilistic ?ood hazard maps for a ?ood prone area indicating the uncertainty bounds of spatial inundation characteristics.A more detailed description of the IHAM modelling system is provided by V orogushyn et al.(2010).
In this paper,the IHAM model has been extended to anal-yse the effect of the uncertainty related to ?ood waveform and to downstream boundary conditions (rating curves)on dike and ?ood hazard mapping.It was set up for the study area of a 50km reach of the Po River between the gauges at Piacenza and Cremona (see Fig.1).
Hydrol.Earth Syst.Sci.,17,3127–3140,2013
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Fig.3.Rating line),and corresponding(Domeneghetti et al.,2012).
2.1
Several
quency analysis plays a dominant role in the overall?ood hazard uncertainty(see e.g.Apel et al.,2008;Merz and Thieken,2009).In particular,an appropriate estimation of peak discharge and?ood volume associated with a speci?c return period is important when?ood hazard is related to dike stability(V orogushyn et al.,2009).For piping and slope-instability,the peak water level and also the duration of dike impoundment,which is related to?ood volume,are decisive. The shape of the?ood event and the duration of high wa-ter levels in the river could strongly in?uence dike-breaching mechanisms,activating piping or micro-instability phenom-ena which may not be observed for a high peak and low vol-ume event.Furthermore,even in the case of a dike breach due to overtopping,the shape of the?ood event and its over-all?ood volume could in?uence the over?ow volume,and consequently the inundated area.In light of these considera-tions,the uncertainty in?ood event estimation considering both?ood peak and volume is addressed adopting differ-ent?ow hydrographs as upstream boundary conditions in a Monte Carlo framework(see Sect.3.4).
2.2Uncertainty in downstream boundary condition Domeneghetti et al.(2012)proposed a general numeri-cal procedure for quantifying global uncertainty of stage-discharge relationships by using numerical hydrodynamic models.Referring to the Cremona river cross-section(see Fig.1)and considering errors affecting river?ow measure-ments(UNI EN Rule748:1997,1997,ISO748:97),the au-thors applied two different procedures for rating-curve esti-mation,which they termed Traditional and Constrained ap-proach,and they quanti?ed the global uncertainty for both (Fig.3).Grey dots in Fig.3represent stage-discharge points simulated by means of a quasi-2-D model of the River Po that has Cremona as an internal cross-section(the downstream boundary condition in this model is set300km downstream).
for a spe-
the hydraulic
10historical ?ood events.The compound of discharge-level pairs simu-lated at the Cremona gauge(grey dots in Fig.3)were then used to mimic several synthetic?eld-measurements cam-paigns(each one of which were made up of15discharge-stage pairs,for details see Domeneghetti et al.,2012).The Traditional approach constructs a rating curve by?tting a series of stage-discharge values observed within the range of measurable stream?ows(i.e.6000m3s?1at Cremona, EU ISO EN Rule1100-2:2010,2010,ISO1100-2:10),while the Constrained approach refers to one additional stage-discharge pair computed by means of a simple1-D steady-state model that also uses Cremona as an internal cross-section.The1-D model is?rst calibrated referring to the maximum measured pair of each synthetic campaign and then used to estimate the maximum discharge capacity at the Cremona section.The Constrained rating curve is?nally es-timated by?tting measured discharge and water-level pairs and by concurrently forcing the curve to honour the esti-mated maximum discharge capacity of the Cremona cross-section(Domeneghetti et al.,2012).The reduction in extrap-olation errors ensured by the Constrained approach,which is visible in Fig.3,results in reduced bias and variability of the estimated rating curves.
Repeating the procedure for several synthetic?eld-measurement campaigns,(see Domeneghetti et al.,2012, for details)the median(red dashed line in Fig.3)and the 90%con?dence interval(thin black lines in Fig.3)for both methodologies were estimated.In particular,left and right panels of Fig.3report the“true”or reference normal rat-ing curve(blue thick line)obtained at Cremona river cross-section from the compound of unsteady stage-discharge pairs (grey dots).Also,the left panel of Fig.3reports the global uncertainty relative to the Traditional approach.In this case, the extrapolation error associated with the utilisation of the curve beyond the range of observed data introduces a
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signi?cant deviation with respect to the reference normal rat-ing curve,as it is clearly illustrated by the width of90% con?dence interval and bias of the Traditional rating curve in Fig.3.The right panel of Fig.3reports the median rating curve(red dashed line)and90%con?dence interval relative to the Constrained approach.
Our study analyses the impacts of rating-curve uncer-tainty on?ood hazard mapping and highlight the differences between Traditional and Constrained approaches to rating-curve construction,comparing them with the results that one would obtain by using a single deterministic median rating curve.
3Study area and model implementation
Our study considers a50km reach of the middle-lower por-tion of the Po River(Fig.1),which spans from Piacenza(up-stream gauge)to Cremona(downstream gauge).The reach can be characterised as a unicursal river,having a width vary-ing between200and500m and a wide?oodplain area.The ?oodplain inside the major river embankments is partly cul-tivated and plots are additionally protected by a system of minor dikes(Castellarin et al.,2011).
3.11-D Model
The hydrodynamic simulation of the?ood wave propaga-tion along the study reach is carried out using a1-D model based on the full Saint-Venant equations numerically solved with the classical implicit four-point?nite difference scheme (Wilson Engineering,2003).The channel geometry is char-acterised by29cross-sections(Fig.1)derived from a2m DTM recently provided by AdB-Po(2005),which combine information collected by means of LiDAR(data collected using two different laser scanners:3033Optech ALTM and Toposys Falcon II),multi-beam sonar survey for the naviga-ble portion of the river and data retrieved by means of tradi-tional ground survey of river cross-sections.
The cross-sections are extracted from the DTM follow-ing the rules for optimal cross-section spacing(Castellarin et al.,2009).The unsteady1-D model is driven by a?ow hy-drograph and conditioned through a rating curve as a down-stream boundary.The representation of tributaries is limited to the River Adda,which is the biggest along the considered reach of the Po River.The Adda contribution is modelled as a lateral in?ow hydrograph as the tributary may appreciably al-ter the Po stream?ow downstream of its mouth.Considering their negligible contributions during the major?oods events experienced along the study reach in the1994and2000the Nure and Chiavenna streams are not considered as tributaries during?ood simulations.
The1-D model is calibrated for a?ood event with an esti-mated return period of approximately50yr,which occurred in the Po River in October2000.The October2000event reproduces the hydraulic behaviour of the study reach in case of extreme?oods because all?oodplains protected by the system of minor dikes were?ooded during the event.The1-D model is calibrated by manually adjusting the roughness coef?cients to match the maximum water levels that were provided by the wrack marks along the reach.The model calibration is performed twice:(1)adopting the Traditional median rating curve(Fig.3,red dashed line of the left panel) and(2)using the Constrained median relation(red dashed line on the right panel of Fig.3).
The high water marks of October2000?ood are accu-rately reproduced by the model,with a mean squared error (MSE)of0.22and0.28m for the Constrained and the Tradi-tional case,respectively.MSE values are not negligible,but they may be regarded as satisfactory due to the magnitude of the simulated?ood event and simpli?cations adopted in the geometrical description of the riverbed(pure-1-D model and single roughness coef?cient for main channel and lateral ?oodplains).
Calibrated Manning’s values mainly vary between0.04 and0.05,and therefore they are in good agreement with those estimated by previous studies on the same reach(see e.g.Castellarin et al.,2009,2011;Domeneghetti et al.,2012; Di Baldassarre and Montanari,2009).
3.2Dike breach module
The main embankment system was discretised into several sections,each one with a length of about1.2km resulting in 28and32sections,respectively,for the right and left side of the embankment system(Fig.1,lower left panel).Dur-ing the simulation,each section is tested for dike stability us-ing fragility functions,which provide the probability of dike-section failure upon hydraulic loading simulated by the1-D hydrodynamic model.Fragility functions for each breach mechanism(overtopping,piping and micro-instability)were developed for each dike section based on the geotechnical and geophysical characteristics of the embankment system, which were compiled by the River Po Basin Authority(AdB-Po-GEOVIT,2004;AdB-Po-DISEG,2001)or derived from the literature and summarised by V orogushyn et al.(2010). In the case of dike failure,breach width(B w)is stochasti-cally sampled through a Monte Carlo procedure from a trun-cated log-normal probability density function?tted to a se-ries of historical observations in the Po river system(see Fig.4,Table2and Coratza,2005).The truncated distri-bution is constrained by the minimum and maximum val-ues of(B w)observed in the Po River system(see Table2). This probabilistic approach was adopted as an alternative to physically based morphodynamic modelling in order to ad-dress the uncertainty associated with the estimation of ul-timate breach widths.Breach width morphodynamic mod-elling remains highly uncertain(Wahl,2001)and no simple and robust relationships between ultimate breach widths and hydrologic and morphologic parameters of?oodplain areas
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Fig.4.Empirical frequency distribution of breach widths,B w,ob-served along the Po River in the period1800–1951(bars)and?tted probability density distribution(blue line;log-normal).
that control the breach through?ow have been developed so far.
The breach development time(h w)was adopted in the range0.5–4h and assumed to follow a normal distribution with mean of2and standard deviation of1.5h.The resulting values for breach times are comparable with those adopted in other studies conducted for the same or comparable rivers (e.g.Apel et al.,2004;Alkema and Middelkoop,2005;Di Baldassarre et al.,2009;V orogushyn et al.,2010;Han et al., 1998).
3.32-D model
In the case of a dike failure,the?ood propagation over the dike-protected?oodplains is simulated by a2-D raster-based model run on a50m×50m resolution grid.The topograph-ical information for the whole study area(Fig.1;global ex-tension890km2)were retrieved from the ASTER GDEM (Advanced Spaceborne Thermal Emission and Re?ection Radiometer–Global Digital Elevation Model;www.gdem. aster.ersdac.or.jp)and rescaled to the coarser grid resolution in order to reduce the computational load.
Considering the absence of detailed information on inun-dation extents experienced in the area of interest and to the uniqueness of breach event,the calibration of the2-D raster-based model appeared to be a dif?cult task.Consequently, spatially distributed Manning’s roughness coef?cients were assigned to each cell based on literature values(Chow,1959) for land use classes retrieved from CORINE land use classi-?cations(COoRdination of Information on the Environment –Land Cover,2006).Table2.Width of dike breaches,B w:statistics observed along the Po River in the period1800–1951(data from Coratza,2005)
B w statistics for
the Po River Obs.value
Number of historical breaches
with observed B w84
Mean B w[m]240
Median B w[m]180
Min B w[m]27
Max B w[m]1200
3.4Development of?ood scenarios and model
simulations
In order to account for the?ood volume,which can be relevant for the stability of?ood protection structures (V orogushyn et al.,2009;Klein et al.,2010),we applied a copula-based bivariate?ood frequency analysis.
The annual maximum peak discharge(q)and the corre-sponding?ood volume(v)observed in a time window of30 days around the?ood peak(10days before the peak,rising limb,and20days after the peak,recession limb)were ex-tracted from the mean daily?ow series in the period from 1951to2008at gauge Piacenza.The adopted time span of 30days entirely embraces the?ood waves that occurred at the study reach;it is evidently site-speci?c and should be re-considered in other case studies.The dependence structure of the couple of variables(Q,V)was described using a copula approach.Among several?tted copulas,the Gumbel copula provided the best?t to the empirical relationship between Q and V according to the selected criteria(i.e.RMSE,AIC, Kolmogorov–Smirnov test and tests based on the empirical copula and on Kendall’s transform(Genest et al.,2009;Fer-manian,2005).
Indicated as F Q(q)and F V(v)the marginal distribution functions of Q and V(a GEV and a log-normal distribution, respectively),the relationship between the uniformly dis-tributed variables u=F Q(q)and v=F V(v)can be expressed by means of the Gumbel copula(1)
Cθ(u,v)=e?[(?ln u)θ+(ln v)θ]1/θ,(1) whereθ 1is a dependence parameter estimated over the set of observations(Salvadori and De Michele,2007). Figure5illustrates the selected?ood events associated with different return periods,Tr.A critical event is deter-mined if either Q or V exceeds given thresholds de?ned through the copula function associated with an exceedance probability(“OR”-case).We focused on a return period of 200yr(hereafter also referred to as Tr200),which is the ref-erence recurrence period adopted by AdB-Po for designing and verifying the main embankment system of the Po River. Red dots of Fig.5a indicate the(q,v)pairs used to discretise the Tr200contour line in our study,by means of which we
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Fig.5.Bivariate analysis:(a)level curves for the Gumbel copula for different return periods(black lines)and(q,v)pairs adopted for the 200yr event(red dots);(b)?ow hydrographs corresponding to copula-based(Q,V)pairs.
took into account the natural variability of?ood hydrographs. These events are not equivalent in terms of?ood volume and peak discharges and they also differ in terms of their prob-ability of occurrence(V olpi and Fiori,2012).We assigned to each one of the?ve selected(q,v)pairs(or scenarios)a probability of occurrence estimated as follows:we computed the joint probability density function(joint pdf)along the contour line based on marginal probabilities(see e.g.V olpi and Fiori,2012);the contour line has been discretised into ?ve stretches,which were identi?ed by halving the curvilin-ear distance between two scenarios;?nally,the relative fre-quency of occurrence of each scenario was estimated as the integral of the joint pdf over each stretch and standardised by the integral of the joint pdf over the entire level curve(see legend of Fig.5b).
We retrieved the shape of the synthetic?ow hydrographs analysing the series of historical?ood events recorded at Pia-cenza.Estimated base?ow was?rst subtracted from each ob-served hydrograph,which was then divided by the maximum discharge,obtaining a dimensionless hydrograph with unit peak?ow.We then computed the mean of all dimensionless ?ood hydrographs and rescaled the resulting mean hydro-graph to match peak discharges and?ood volumes estimated through the bivariate analysis for the?ve Tr200events(red dots in Fig.5a).Figure5b reports the?ve synthetic hydro-graphs obtained in the study and their corresponding empir-ical relative probability of occurrence.IHAM was driven by the developed?ood hydrographs,taking into account the rel-ative probability of each.To investigate the effect of rating-curve uncertainty on?ood hazard estimation,?ood scenar-ios were simulated,adopting different downstream boundary conditions de?ned for the Cremona gauge.Approximately 8000Monte Carlo simulations were run in total to propagate the uncertainty in upstream and downstream boundary con-ditions to?ood hazard estimations.In particular,subsets of ～2000runs were used to explore the effects of uncertainty on?ood hazard mapping:
–MedianT subset;?ow hydrographs were randomly se-lected as upstream boundary conditions,whereas the median rating curve for Traditional approach(red dashed lines on left panel of Fig.3)was used as down-stream boundary conditions.
–MedianC subset;same as before but adopting the Con-strained median rating curve as downstream boundary condition.
–RandomT subset;both upstream(i.e.?ow hydrographs) and downstream(i.e.Traditional rating curves)bound-ary conditions are stochastically sampled.In particular, referring to the left panel of Fig.3,the rating curve is sampled between the90%con?dence interval(black lines in the Figure)during each Monte Carlo simula-tion.
–RandomC subset;same as before by considering Con-strained rating curves.
4Results
Figure6reports results provided by the1-D model for the RandomT subset.The upper panel of the?gure reports the minimum levee-crest elevation(red dashed line)for the study reach of the Po River and compares it with the median(black line)and the range of variability(grey dashed lines)of wa-ter surface simulated for the Tr200event.In the lower panel of Fig.6,the water depth variability simulated for RandomT (black line)is compared with the one obtained from the Ran-domC subset(dashed line).The lower panel of Fig.6clearly shows the impact of rating-curve uncertainty in terms of wa-ter levels along the downstream end of the studied Po River reach(RandomT and RandomC subsets).The variability in-troduced by the downstream boundary condition in?uences the water levels simulated upstream through a backwater ef-fect for a remarkable distance(i.e.25–35km in this case).
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Fig.6.Monte Carlo simulations;upper panel:range of variation(grey dashed line)and median(black line)water elevation pro?les simulated for the RandomT subset along the Po River,compared with the minimum dike-crest elevation(red dashed line).Lower panel:water depth variability simulated along the Po River for RandomT(black line)and RandomC(dashed line)subsets.
Panel(a)of Fig.7reports the probabilistic?ood hazard map related to a speci?c return period of200yr and obtained for the MedianT subset(～2000runs);the probability of in-undation of each cell in the?ood prone area is indicated through a blue colour scale.Such a measure is calculated as the ratio between the number of simulations in which the cell is wet(i.e.water depth>0cm)and the total number of Monte Carlo runs.
High probabilities of inundation in Fig.7a are symp-tomatic of the presence of critical river stretches(e.g.higher probability of overtopping).Although the?ooding probabil-ities for the majority of the areas are quite small ranging be-tween20–30%,results highlight a critical condition in the embankment stretch located downstream of Torrente Chi-avenna tributary.In this case,a local depression on the dike crest results in a high probability of overtopping,leading to a remarkable probability of inundation for the?ood-prone area opposite to the tributary mouth(dark blue colour in Fig.7a). The probabilistic map in Fig.7b reports the difference in probability of inundation arising from the consideration of the uncertainty bounds around the median Traditional rating curve,that is the difference between the inundation probability obtained for the MedianT and RandomT sub-sets.Figure8a shows the probabilistic inundation map ob-tained for the MedianC subset(～2000Monte Carlo runs), while Fig.8b provides the difference in inundation proba-bility between probability inundation maps obtained for the MedianC and RandomC subsets(～2000Monte Carlo runs each).
The map reported in Fig.8b does not highlight tangible variations in the probability of?ooding for the area outside the main embankments due to the rating-curve uncertainty. Patchy variations(shown in Fig.8b)seem to be caused by the stochastic de?nition of breach dimension and develop-ment time.Concerning the rating-curve uncertainty,relative to the unbiased rating curve constructed with the Constrained approach,Fig.8clearly shows the importance of the variabil-ity of the rating curve,i.e.con?dence interval width(lower panel of Fig.6).In this case,the reduced uncertainty,i.e. the narrow con?dence interval and small extrapolation errors (right panel of Fig.3),results in a limited effect on?ood es-timation and inundation assessment.
Figure9compares probabilistic?ood hazard maps com-puted on the basis of the RandomT,panel(a),and Ran-domC,panel(b),subsets(～2000Monte Carlo runs each). Areas highlighted in the?gure emphasise the difference be-tween the two subsets.Even though the uncertainty in down-stream boundary conditions is considered in both scenarios, the highlighted area appears to be inundated only when the Traditional approach is considered.This result could have been expected in light of the evident extrapolation error af-fecting rating curves constructed with the Traditional ap-proach,which results in a worsening of the overtopping phe-nomenon.
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Fig.
Fig.
Fig.(b:
In all considered scenarios,overtopping was the only breach mechanism responsible for dike failure.None of the dike sections failed due to piping and micro-instability,even though the bivariate approach ensured a signi?cant variabil-ity of the?ood volume for the200yr?ood event(see Fig.5). 5Discussion
It is worth emphasising here that,when assessing the?ood hazard conditions along a river reach dominated by sub-critical?ow conditions,extending the modelling domain far enough downstream the stretch of interest may limit(or com-pletely remove)the possible negative effects of uncertain downstream boundary conditions.Nevertheless,we deem the assessment of the effects of uncertainty from downstream boundary conditions on?ood hazard mapping to be very important for two main reasons.First,the extension of the modelling domain could be costly,or hampered by various practical limitations(lack of topographical data,computa-tional burden,time constraints,etc.).Second,the backwater length that is needed for identifying the optimal location of the downstream boundary condition is generally estimated through simpli?ed computational schemes and,hence,the backwater length is rather uncertain too.
For instance,for subcritical?ow conditions with a Froude number signi?cantly lower than one,Samuels(1989)sug-gests computing the distance where the backwater upstream of a control(as well as other disturbances)decays to less than 0.1of the original value, x,as
x≈0.2D/s,(2)
where D is the bankfull depth of?ow and s is the surface(or main channel)slope.For the considered river reach,with a bankfull depth D of～16m and an average channel slope s of0.25‰,Eq.(2)returns～https://www.wendangku.net/doc/fa5632203.html,ing a1-D hydraulic model of the study reach to calculate the steady-state wa-ter surface pro?le relative to a discharge of12500m3s?1, and setting the water depth1.5m above the normal depth at the downstream end to mimic the uncertainty associ-ated with Traditional rating curve(see left panel of Fig.3), one may easily verify that x(de?ned above)results equal to～18km.It should be noted that both estimates are signif-icantly lower than25–35km,that is the distance at which the variability introduced by the downstream boundary condition in?uences the simulated inundation probabilities through a backwater effect.This result,obtained from a MC simu-lation experiment and the cascade of numerical hydrody-namic models,clearly highlights that simpli?ed modelling approaches,as those brie?y recalled above,may signi?cantly underestimate the effects of the uncertainty in downstream boundary conditions.
Results presented in the previous section clearly high-light the remarkable impact of the methodology applied for rating-curve construction and associated uncertainty on?ood hazard assessment,and in particular on dike breaching and inundation probability.The variability of rating curves pro-duces a signi?cant uncertainty in?ood probability estima-tion.Figure7b shows for our case study that the determinis-tic(i.e.neglecting uncertainty,MedianT subset)utilisation of a rating curve constructed using a traditional approach(i.e.?tting the available discharge-water level observations)re-sults in a signi?cant underestimation of?ooding probability. As a general remark,it is worth noting here that the?ood-ing probability could be underestimated or overestimated in other study areas depending on local conditions,yet we want to underline the signi?cant bias that may affect the?ood in-undation estimates.The bias sign depends on the speci?c lo-cal conditions,which may produce systematic underestima-tion,or overestimation,of the water levels in the downstream end of the considered river reach,affecting the overtopping probabilities of the main embankments.
The limited variability of rating curves obtained by means of Constrained approach entails a reduced variability in terms of water elevation along the river and this results in a more re-liable evaluation of dike stability and likelihood of?ooding. Although we are aware that this result could also be partly associated with our case-study,the analysis reveals how the reduction of the extrapolation error could be a good strat-egy in order to reduce bias and uncertainty on?ood hazard estimation when the uncertainty of the rating curve cannot be considered(i.e.deterministic interpretation of the curve) and has to be neglected during?ood hazard assessments for various practical limitations(e.g.when performing real-time ?ood inundation modelling).
The two maps in Fig.9emphasise the effects on inunda-tion probability estimates of bias on water levels that might be associated with a Traditional approach to rating-curve construction relative to the Constrained approach(see also Fig.3).The comparison of these maps highlights a possible misinterpretation of hazard estimation due to extrapolation errors associated with the curve?tting exercise(i.e.rating-curve construction).The highlighted cells(red ellipse)ap-pear?ooded only in the case of the application of the Tra-ditional approach(Fig.9a),and this is a consequence of the better reproduction of the hydraulic behaviour of the Po River at Cremona cross-section ensured by the Constrained rating curve(see Fig.3).
Concerning the scienti?c debate on probabilistic versus deterministic inundation maps,some considerations may be raised from Fig.8b which illustrates the difference in terms of inundation probability between RandomC and MedianC subsets.Both subsets refer to the Constrained approach for constructing rating curves,therefore the most accurate of all considered cases both in terms of possible extrapolation er-rors(i.e.limited bias)and global rating-curve uncertainty (i.e.limited con?dence interval).The comparison between the two maps enables one to understand the difference in terms of?ood probability that originates from the uncer-tainty in the downstream boundary condition.Differences,
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although limited in terms of inundation probability,are present over wide regions(see for instance the green area in Fig.8b)and,more importantly,they are located quite far from the gauged river cross-section where the boundary con-dition is set(～25–35km upstream of the downstream end). Furthermore,it should be emphasised that even though these differences in terms of?ooding probability are modest they may result in a non-negligible variation in terms of overall ?ood risk,e.g.due to the interactions of hazard(probability) with exposure and vulnerability of the area,especially when a wider spectrum of return periods is considered(as opposed to Tr=200yr selected in this study).
Similar results in terms of overall extension of the?ood-able area may also be obtained from deterministic inundation maps,however,the added value of probabilistic inundation mapping relies on its capability to represent the uncertainty of the output in a very effective way.The representation of the uncertainty associated with the output facilitates the in-teraction between scientists and decision-makers,who may or may not have a strong background in numerical-hydraulic modelling.
To promote the utilisation of probabilistic maps,scientists should provide decision-makers with information on their meaning and stress the distinction between the overall in-undation probabilities and those related to a speci?c return period.Inundation probabilities represented in this work(see e.g.Sect.4and Figs.7and9)refer to a200yr return pe-riod and are therefore speci?c to a particular set of syn-thetic?ood events.The overall probability of inundation for a given point within the?ood-prone area of the River Po could be different if all possible?ood return periods and re-spective scenarios were considered.However,since the Po River Basin Authority grounds the?ood risk assessment, management and mitigation by considering a speci?c re-turn period,i.e.200yr,the probability of inundation asso-ciated with200yr?oods is a meaningful representation of the?ood hazard for the decision-making process.Once ade-quately informed,the decision-maker will decide how best to deal with this uncertainty(e.g.by including highlighted ar-eas of Fig.9b among the restricted areas in the spatial plan-ning acts)and weight her/his decision by the probability of ?ooding.On the other hand,if the decision on the appropriate protection level is required based on the overall?ood risk as-sessment,the entire spectrum of the return periods should be taken into account with their respective uncertainty estima-tions.This is also feasible with the presented methodology. Finally,concerning our particular case study,the analy-sis pointed out that dike stability is strongly controlled by peak discharges rather than by?ood volumes.Although the variability of?ood volume was explicitly considered in the ?ood hydrograph scenarios(Fig.5),the embankment sys-tem was in fact found to be sensitive to overtopping fail-ures only.Failures due to piping or micro-instability did not occur in the Monte Carlo runs due to the remarkable thick-ness of the main river embankments(average riverside slope 1:2or1:3;average landside slope1:5–1:6).This result is in agreement with what has been observed along the study reach of the Po River during the October2000?ood event (Coratza,2005).However,evidences of sandboils along the study reach during recent?ood events in1994(magnitude similar to the October2000event)and2000suggest a start-ing retrogressive erosion and the presence of a non-negligible danger of piping(Coratza,2005).Therefore,breach mecha-nisms other than overtopping should not be excluded from the analysis a priori.
6Conclusions
The debate relative to the deterministic and probabilistic ap-proach for?ood hazard estimation is still ongoing in the scienti?c community(Di Baldassarre,2012;Di Baldassarre et al.,2010;Montanari,2007).Providing?ood probability maps for the?ood prone areas appears to be an ef?cient way to visualise the likelihood of?ooding and it also offers addi-tional information concerning the reliability of its estimation. The scienti?c community is well aware of all risks asso-ciated with deterministic statements(i.e.binary,yes or no kind of statements)when the system under study is uncertain. Nevertheless,the output of numerical simulations as well as hydraulic and hydrological input data are often used in prac-tice and applied regardless of their uncertainty.Probabilis-tic inundation maps are still scarcely adopted as additional assets by decision-makers called to de?ne?ood protection strategies.This should mainly be attributed to a lack of spe-ci?c guidelines as well as to a limited ability of the scienti?c community to communicate the meaningfulness and effec-tiveness of this kind of spatial representation of?ood haz-ard.We investigated the effects of the uncertainty in the def-inition of the downstream boundary condition given by the rating curve on the?ood probability estimation for a diked reach of the Po River.The evaluation was carried out with the IHAM model,which enables the evaluation of failure prob-abilities of the dike system under variable hydraulic condi-tions and for different breaching mechanisms.The intrinsic uncertainty in?ood hydrographs was considered through a bivariate approach by modelling the correlation structure of peak stream?ow and?ood volume by means of a copula ap-proach.
Results of the analysis highlight how rating curves’un-certainty signi?cantly affects?ood mapping assessment and, in particular,probabilistic?ood mapping,when the curves themselves are used as downstream boundary conditions. This aspect appears particularly relevant when the range of uncertainty for high?ow rates becomes wide due to the extrapolation error introduced during rating-curve construc-tion.In this context,the methodology used for rating-curve construction plays a fundamental role in the model chain for?ood hazard assessment.We investigated the effects in terms of dike breaching and inundation probability of two
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methodologies for rating-curve construction,referred to in the study as Traditional and Constrained approaches(see also Domeneghetti et al.,2012).In the case of rating curve con-structed by means of a typical approach(e.g.Traditional ap-proach)the analysis shows through a series of Monte Carlo simulation experiments that neglecting the uncertainty as-sociated with empirical rating curve may lead to highly in-accurate,and therefore dangerous,inundation mapping.In this context,the study clearly points out how taking into ac-count the rating-curve uncertainty through a probabilistic ap-proach signi?cantly enhances the reliability of the?ood haz-ard mapping.Also,the results of our analysis pointed out that limiting the extrapolation error while constructing em-pirical rating curves(for instance by adopting an approach similar to the so-call Constrained approach illustrated in Domeneghetti et al.,2012)signi?cantly reduces the effect of uncertain boundary conditions on the?ood likelihood esti-mation.Additionally,the reduction of rating-curve bias leads to a more reliable?ood hazard estimation,reducing the risk of unfounded estimation of?oodable areas.This is an impor-tant aspect when practical constraints(https://www.wendangku.net/doc/fa5632203.html,ck of data,avail-able time,money,etc.)prevent the modeller from extending the study domain downstream,i.e.locating the downstream boundary conditions suf?ciently far enough away from the area of interest,which could evidently reduce,or completely remove,the effect of rating-curve uncertainty on model re-sults.A probabilistic statement of?ood hazard,which in-corporates a quanti?cation of the uncertainty affecting the output of numerical hydraulic modelling,represents a funda-mental piece of information for decision-makers,when,for instance,they are called to de?ne spatial development plans for a given area,or when they need to identify priorities in the organisation of civil protection actions during a?ood event. Probabilistic?ood inundation maps are the most natural and straightforward graphical representation of such a statement, and should always be preferred to deterministic inundation maps.
Acknowledgements.The authors are extremely grateful to the Interregional Agency for the Po River(Agenzia Interregionale per il Fiume Po,AIPO,Italy)and Po River Basin Authority(Autorit′a di Bacino del Fiume Po,Italy)allowing access to their high-resolution DTM of River Po.We are also grateful to an anonymous reviewer, to Micha Werner and to the Editor,Thom Bogaard,for their thorough and constructive reviews.
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